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Properties

Label 4-1350e2-1.1-c1e2-0-17
Degree 44
Conductor 18225001822500
Sign 11
Analytic cond. 116.204116.204
Root an. cond. 3.283263.28326
Motivic weight 11
Arithmetic yes
Rational yes
Primitive no
Self-dual yes
Analytic rank 00

Origins

Origins of factors

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Normalization:  

Dirichlet series

L(s)  = 1  + 2-s + 7-s − 8-s − 2·11-s + 6·13-s + 14-s − 16-s + 4·17-s + 12·19-s − 2·22-s + 23-s + 6·26-s + 9·29-s + 2·31-s + 4·34-s + 4·37-s + 12·38-s − 11·41-s + 4·43-s + 46-s − 7·47-s + 7·49-s − 56-s + 9·58-s − 4·59-s + 7·61-s + 2·62-s + ⋯
L(s)  = 1  + 0.707·2-s + 0.377·7-s − 0.353·8-s − 0.603·11-s + 1.66·13-s + 0.267·14-s − 1/4·16-s + 0.970·17-s + 2.75·19-s − 0.426·22-s + 0.208·23-s + 1.17·26-s + 1.67·29-s + 0.359·31-s + 0.685·34-s + 0.657·37-s + 1.94·38-s − 1.71·41-s + 0.609·43-s + 0.147·46-s − 1.02·47-s + 49-s − 0.133·56-s + 1.18·58-s − 0.520·59-s + 0.896·61-s + 0.254·62-s + ⋯

Functional equation

Λ(s)=(1822500s/2 ΓC(s)2 L(s)=( Λ(2−s)\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}
Λ(s)=(1822500s/2 ΓC(s+1/2)2 L(s)=( Λ(1−s)\begin{aligned}\Lambda(s)=\mathstrut & 1822500 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}

Invariants

Degree: 44
Conductor: 18225001822500    =    22⋅36⋅542^{2} \cdot 3^{6} \cdot 5^{4}
Sign: 11
Analytic conductor: 116.204116.204
Root analytic conductor: 3.283263.28326
Motivic weight: 11
Rational: yes
Arithmetic: yes
Character: Trivial
Primitive: no
Self-dual: yes
Analytic rank: 00
Selberg data: (4, 1822500, ( :1/2,1/2), 1)(4,\ 1822500,\ (\ :1/2, 1/2),\ 1)

Particular Values

L(1)L(1) ≈\approx 4.4186350804.418635080
L(12)L(\frac12) ≈\approx 4.4186350804.418635080
L(32)L(\frac{3}{2}) not available
L(1)L(1) not available

Euler product

   L(s)=∏pFp(p−s)−1L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1}
ppGal⁡(Fp)\Gal(F_p)Fp(T)F_p(T)
bad2C2C_2 1−T+T2 1 - T + T^{2}
3 1 1
5 1 1
good7C2C_2 (1−5T+pT2)(1+4T+pT2) ( 1 - 5 T + p T^{2} )( 1 + 4 T + p T^{2} )
11C22C_2^2 1+2T−7T2+2pT3+p2T4 1 + 2 T - 7 T^{2} + 2 p T^{3} + p^{2} T^{4}
13C22C_2^2 1−6T+23T2−6pT3+p2T4 1 - 6 T + 23 T^{2} - 6 p T^{3} + p^{2} T^{4}
17C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
19C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
23C22C_2^2 1−T−22T2−pT3+p2T4 1 - T - 22 T^{2} - p T^{3} + p^{2} T^{4}
29C22C_2^2 1−9T+52T2−9pT3+p2T4 1 - 9 T + 52 T^{2} - 9 p T^{3} + p^{2} T^{4}
31C22C_2^2 1−2T−27T2−2pT3+p2T4 1 - 2 T - 27 T^{2} - 2 p T^{3} + p^{2} T^{4}
37C2C_2 (1−2T+pT2)2 ( 1 - 2 T + p T^{2} )^{2}
41C22C_2^2 1+11T+80T2+11pT3+p2T4 1 + 11 T + 80 T^{2} + 11 p T^{3} + p^{2} T^{4}
43C22C_2^2 1−4T−27T2−4pT3+p2T4 1 - 4 T - 27 T^{2} - 4 p T^{3} + p^{2} T^{4}
47C22C_2^2 1+7T+2T2+7pT3+p2T4 1 + 7 T + 2 T^{2} + 7 p T^{3} + p^{2} T^{4}
53C2C_2 (1+pT2)2 ( 1 + p T^{2} )^{2}
59C22C_2^2 1+4T−43T2+4pT3+p2T4 1 + 4 T - 43 T^{2} + 4 p T^{3} + p^{2} T^{4}
61C22C_2^2 1−7T−12T2−7pT3+p2T4 1 - 7 T - 12 T^{2} - 7 p T^{3} + p^{2} T^{4}
67C2C_2 (1−16T+pT2)(1+5T+pT2) ( 1 - 16 T + p T^{2} )( 1 + 5 T + p T^{2} )
71C2C_2 (1−6T+pT2)2 ( 1 - 6 T + p T^{2} )^{2}
73C2C_2 (1+4T+pT2)2 ( 1 + 4 T + p T^{2} )^{2}
79C22C_2^2 1−12T+65T2−12pT3+p2T4 1 - 12 T + 65 T^{2} - 12 p T^{3} + p^{2} T^{4}
83C22C_2^2 1+11T+38T2+11pT3+p2T4 1 + 11 T + 38 T^{2} + 11 p T^{3} + p^{2} T^{4}
89C2C_2 (1+T+pT2)2 ( 1 + T + p T^{2} )^{2}
97C22C_2^2 1−8T−33T2−8pT3+p2T4 1 - 8 T - 33 T^{2} - 8 p T^{3} + p^{2} T^{4}
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   L(s)=∏p ∏j=14(1−αj,p p−s)−1L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}

Imaginary part of the first few zeros on the critical line

−9.661551721970156061428062779576, −9.652709068434193011728793665670, −8.904348762536853763362943191910, −8.534239114370644041235166018900, −8.153927211192121552050246523868, −7.899356743076805658758081521360, −7.28249958751412230256077157485, −7.03641321811563206808671022494, −6.26795048789240915354021246679, −6.12705299990550386674174035839, −5.49168491834556954928837097563, −5.13509841862378702042054862255, −4.94421653299149623475874815360, −4.28798078596763913223665098551, −3.49539953319052853498205150878, −3.46938548816308408760051880023, −2.93027409699647923000437327128, −2.23833209356571198125667153809, −1.11919988156753806813512814014, −1.03038065239971203891007142409, 1.03038065239971203891007142409, 1.11919988156753806813512814014, 2.23833209356571198125667153809, 2.93027409699647923000437327128, 3.46938548816308408760051880023, 3.49539953319052853498205150878, 4.28798078596763913223665098551, 4.94421653299149623475874815360, 5.13509841862378702042054862255, 5.49168491834556954928837097563, 6.12705299990550386674174035839, 6.26795048789240915354021246679, 7.03641321811563206808671022494, 7.28249958751412230256077157485, 7.899356743076805658758081521360, 8.153927211192121552050246523868, 8.534239114370644041235166018900, 8.904348762536853763362943191910, 9.652709068434193011728793665670, 9.661551721970156061428062779576

Graph of the ZZ-function along the critical line